5.76 Lecture #33 5/08/91 Page 1 of 10 pages. Lecture #33: Vibronic Coupling

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1 5.76 Lecture #33 5/8/9 Page of pages Lecture #33: Vibroic Couplig Last time: H CO A A X A Electroically forbidde if A -state is plaar vibroically allowed to alterate v if A -state is plaar iertial defect says A -state is ot plaar expect to see all v if ot plaar staggerig of v level spacigs iversio through low barrier to plaarity dyamic vs. rigid molecule symmetry classificatio: molecular symmetry group How does vibroic couplig really work? What are the vibratioal itesity factors aalogous to Frack-Codo factors i the case of vibroically allowed rather tha electroically allowed trasitio? See T. Azumi ad K. Matsuzaki, Photochemistry ad Photobiology 5, 35 (977) for readable review article. Outlie: Crude Adiabatic Approximatio Correctio of ψ for effect of eglected off-diagoal matrix elemets H CO A A example What happes to Frack-Codo factors for vibroically allowed trasitio? Two electroic basis sets prediagoalize symmetry-breakig vibroic iteractio Chages i shapes of potetial curves Ies model for vibratioal bad itesities ad level staggerig Recall or-oppeheimer or clamped uclei approximatio. We use this procedure to defie complete sets of electroic ad uclear motio wavefuctios with which we ca FORMALLY expad exact ψ s ad compute (or parametrize) all properties of exact eigestates. The simplest basis set is called CRUDE ADIAATIC (CA) ψ CA ( r,q)=ψ o j ( r,q )χ CA fixed uc lear locatios! ( Q) Q is a coveiet referece structure (usually the equilibrium geometry or a high-symmetry potetial eergy maximum or saddle poit). ψ j o is the electroic wavefuctio i the j-th electroic state computed at the fixed uclear coordiates Q.

2 5.76 Lecture #33 5/8/9 Page of pages χ CA (Q) is the vibratio-rotatio wavefuctio computed from a Schrödiger Equatio. uclear kietic eergy potetial eergy of bare uclei eigevalue of clamped uclei electroic Schrödiger Equatio at Q approximate uclear U(r,Q) = U(r,Q) U(r,Q ) chage i e uclear ad e e Coulomb eergy T N (Q) + V(Q) + o j ( Q )+ ψ o j ( r,q )² U(r,Q) ψ o j r,q ( ) χ CA ( Q) effective potetialeergy surface CA = E χ CA ( Q) Note that the U itegral is evaluated usig effect of distortio of molecule from Q. ψ j o (r, Q ) thus caot cotai the exact We have explicitly excluded the effects of off-diagoal matrix elemets. I order to get a better approximatio to the exact ψ, we must use perturbatio theory to correct ψ. ψ (r,q) =ψ CA (r,q)+ kr =ψ o j (r,q )χ CA (Q) + ψ CA ²Uψ CA { } kr ψ kr E CA CA E kr k j ψ o k (r,q )χ CA kr (Q) r CA (r,q) ( χ CA kr ψ o o CA k ²Uψ j χ ) E CA CA E kr call this a vibroic { } meas itegrate over r ad Q ( ) meas itegrate over Q meas itegrate over r This form of ψ (r,q) is called the Herzberg-Teller expasio. Now expad U(r,Q) i power series about Q i each of the ormal coordiates.

3 5.76 Lecture #33 5/8/9 Page 3 of pages ²U= ² U(r,Q )+ U(r,Q) Q = by defiitio of U Now defie the mixig coefficiet. Q +,m U Q Q m Q Q m etc. γ kr, ( ) U ψ k o r,q Q ψ o j ( r,q ) χ CA CA kt Q χ E CA CA E kt ψ (r,q) =ψ o j (r,q )γ CA (Q) + ut we ca see that γ kr, must vaish if k j r γ kr, γ o k (r,q )χ CA kr (Q) ote vibroic wavefuctio for k-th, NOT j-th electroic state! Γ k Γ j Γ Q OR Γ r Γ t Γ Q which is equivalet to requirig that Γ kr Γ Γ totally symmetric. So ow we are ready to cosider the specific case of the H CO A A state. Out-of-plae edig mode as promoter b A = b vibratio So we are cosiderig vibroic couplig to the state. No-Lecture Let s make a really crude model for the out-of-plae bedig levels of both A ad states. * both are harmoic (N the A state is NOT a double miimum o-plaar state!!) * both have same frequecy ω * couplig is exclusively via U Q term. Q

4 5.76 Lecture #33 5/8/9 Page of pages 3a a b 5a b b b 6a X A ev elect. forbidde b b A A π* 3.5 ev elect. allowed (b-type) 6a b σ* 7. ev elect. allowed (a-type) b b A π* π 8. ev elect. allowed (c-type) b 5a π* σ 9.5 ev A X trasitio ca borrow oscillator stregth by vibroic couplig with via b vibratio because A b = A via a vibratio because A a = A a vibratio via b vibratio because A b = I will ow show that vibroic couplig accouts for both the oscillator stregth for A X trasitio ad the staggerig of ν vibratioal levels i A -state. Assume ν i A ad states is coveiet for harmoic - ot a double miimum o-plaar state calculatig same ω ad ot displaced (ecessary if miimum or maximum is plaar) vibratioal matrix elemets couplig is exclusively via ŽU Q term ŽQ

5 5.76 Lecture #33 5/8/9 Page 5 of pages v = A 3 v = ψ =ψχ + γ ψχ o CA o CA Av A Av v, Av v v o U v Q v o γ, ψ ψ v Av A CA Q E E Av a mass-idepedet electroic factor β A mode #, ot th power µω ( ) / v +δ CA v v, v + + v δ ( ) ² T A ω v v v,v Keepig oly levels of state i Herzberg- Teller expasio

6 5.76 Lecture #33 5/8/9 Page 6 of pages Summary of o-zero matrix elemets 3 v = 3 3 So we have 3 v = lump everythig ito this adjustable costat ψ =ψ χ +βψ v χ + v + χ ψ =ψ χ βψ v χ v + χ o CA o CA CA Av A Av v v + o CA o CA CA v v A Av Av +

7 state 5.76 Lecture #33 5/8/9 Page 7 of pages Trasitio probability for A v X v I = ψ µ ψ Av, X v Av X v µ v ( v ) o o / CA CA / CA CA b X v Xv v + Xv =β ψ ψ χ χ + + χ χ =β M b, X F-C factor vq ( v ) q v ( v ) positive v Xv v + Xv X X v, v v +, v / either sig Note that this is more complicated tha usual FRANCK-CONDON expressio for allowed trasitios. It is expressed i terms of Frack-Codo factors for X NOT A X!!!! We still have a symmetry selectio rule for the ν vibratioal modes because they are o-totally symmetric. From v = we ca oly reach v = eve or A v = odd. Note that the itesity expressio above vaishes for v = ad v = because q X,. To express this more geerally, for ay vibratioal bad i the A X system that is made allowed by vibroic couplig to the promoted by ν. I idividual mode F-C factors A X M q X v Q v,, v v AV XV b X v i vi vi v v symmetry selectio v rule v X = eve b-type v A v = odd v A v X = odd =β K. K. Ies J. Mol. Spectrosc. 99, 9 (983) performed a vibroic couplig calculatio which ot oly reproduced the mode- itesity promotio factors, but also explaied the level staggerig i the A -state.

8 5.76 Lecture #33 5/8/9 Page 8 of pages I order to defie complete basis sets, we solve a approximate Schrödiger equatio by eglectig certai terms i the exact H, or igorig certai off-diagoal elemets of certai terms. I the crude adiabatic approximatio, we defie potetial curves by igorig terms of the form ψ CA CA ² H(r,Q) ψ kr We showed that, by expadig U as power series i Q (the ormal mode displacemets), we get ( ) jk = H electroic ψ j o r,q Q ( ) U ( ) Q = ψ k o r,q γ jk Q We ca go to a ew electroic basis set by diagoalizig H + γ jk Q. Suppose we have two idetical harmoic potetial curves for mode of electroic states j ad k. The we have the followig zero-order ad diagoalized potetial curves. V k (Q ) V k o H =γ jk Q V j o

9 5.76 Lecture #33 5/8/9 Page 9 of pages Upper curve gets arrower. Lower curve turs ito double miimum curve. Q = poits of both curves do ot shift. Vibratioal Eigestates of lower curve will exhibit the patter of a symmetric double miimum potetial. V k o V j o H ij ( Q )=ω k Q ( Q )=ω j Q ( Q )=γ jk Q Secod-order perturbatio theory: ( γ jk ) Q V k =ω k Q + = ω ω ( k ω j )Q k Q +αq + T ek T ej V j = ω j Q αq ( ) ω ( k ω k )= ω ( j ω j )= γ jk T ek T ej opposite sig shifts i harmoic frequecy ( γ jk ) α ω ( T ek T ej ) k ( ω j ) quartic term that depeds o differece i ω's for j ad k. This shows that upper state ω icreases ad lower state ω decreases. Exact treatmet V ± = ω k +ω j ²V E H H ²V ( )Q ²V = ω k ω j Q ± ω ( k ω j ) E Q + γ jk V ± = V k + V j ( ) Q / ± ²V + H For large γ, secod term i [ ] / will domiate at small Q but first term will evetually domiate at large Q. /

10 5.76 Lecture #33 5/8/9 Page of pages A secod-order perturbatio treatmet of this kid of -state iteractio i the CA picture caot give this type of level stagger. It is ecessary to set up ad diagoalize two matrices H I odd quata of upper state eve quata of lower state H II eve quata of upper state odd quata of lower state because of odd-eve symmetry of a symmetric (ot ecessarily harmoic) potetial, there ca be o couplig matrix elemets betwee these two matrices. The level shifts are larger for the lower states i H II tha those i H I. This produces level staggerig. K. K. Ies [J. Mol. Spectrosc. 99, 9-3 (983)] reproduced A X itesity ad A -state level patter with T A A = 835 cm ω =ω = 5 cm, Av v= va A H =βv β= 338 / cm